Classical Analysis: An Approach through Problems

443 Pages·2022·6.156 MB·English

by  Hongwei Chen

Classical Analysis
A conceptually clear induction to fundamental analysis theorems, a tutorial for creative
approaches for solving problems, a collection of modern challenging problems, a pathway to
undergraduate research—all these desires gave life to the pages here.
This book exposes students to stimulating and enlightening proofs and hard problems of
classical analysis mainly published in The American Mathematical Monthly.
The author presents proofs as a form of exploration rather than just a manipulation of symbols.
Drawing on the papers from the Mathematical Association of America’s journals, numerous
conceptually clear proofs are offered. Each proof provides either a novel presentation of a famil-
iar theorem or a lively discussion of a single issue, sometimes with multiple derivations.
The book collects and presents problems to promote creative techniques for problem-solving
and undergraduate research and offers instructors an opportunity to assign these problems as
projects. This book provides a wealth of opportunities for these projects.
Each problem is selected for its natural charm—the connection with an authentic mathemati-
cal experience, its origination from the ingenious work of professionals, develops well-shaped
results of broader interest.
Hongwei Chen received his Ph.D. from North Carolina State University in 1991. He is cur-
rently a professor of mathematics at Christopher Newport University. He has published more
than 60 research papers in analysis and partial differential equations. He also authored Monthly
Problem Gems published by CRC Press and Excursions in Classical Analysis published by the
Mathematical Association of America.
Textbooks in Mathematics
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Introduction to Linear Algebra
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Real Analysis and Foundations, Fifth Edition
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Geometry and Its Applications, Third Edition
Walter J. Meyer
Transition to Advanced Mathematics
Danilo R. Diedrichs and Stephen Lovett
Modeling Change and Uncertainty
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Abstract Algebra
A First Course, Second Edition
Stephen Lovett
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Applied Differential Equations
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Introduction to Computational Mathematics: An Outline
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Mathematical Modeling the Life Sciences
Numerical Recipes in Python and MATLABTM
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Classical Analysis
An Approach through Problems
Hongwei Chen
https://www.routledge.com/Textbooks-in-Mathematics/book-series/CANDHTEXBOOMTH
Classical Analysis
An Approach through Problems
Hongwei Chen
First edition published 2023
by CRC Press
6000 Broken Sound Parkway NW, Suite 300, Boca Raton, FL 33487-2742
and by CRC Press
4 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN
CRC Press is an imprint of Taylor & Francis Group, LLC
© 2023 Hongwei Chen
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Library of Congress Cataloging-in-Publication Data
Names: Chen, Hongwei, author.
Title: Classical analysis : an approach through problems / Hongwei Chen.
Description: First edition. | Boca Raton : CRC Press, 2023. | Includes
bibliographical references and index.
Identifiers: LCCN 2022025390 | ISBN 9781032302478 (hardback) | ISBN
9781032302485 (paperback) | ISBN 9781003304135 (ebook)
Subjects: LCSH: Mathematical analysis--Problems, exercises, etc.
Classification: LCC QA301 .C435 2023 | DDC 515.076--dc23/eng20221007
LC record available at https://lccn.loc.gov/2022025390
ISBN: 978-1-032-30247-8 (hbk)
ISBN: 978-1-032-30248-5 (pbk)
ISBN: 978-1-003-30413-5 (ebk)
DOI: 10.1201/9781003304135
Publisher’s note: This book has been prepared from camera-ready copy provided by the authors.
To Ying, Alex, and Abby,
for whom my love has no upper bound
Taylor & Francis
Taylor & Francis Group
http://taylorandfrancis.com
Contents
Preface ix
1 Sequences 1
1.1 Completeness Theorems for the Real Number System . . . . . . . . . . . . 1
1.2 Stolz-Ces`aro Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.3 Worked Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.3.1 (cid:15)−N definition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.3.2 Cauchy criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.3.3 The squeeze theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.3.4 Monotone convergence theorem . . . . . . . . . . . . . . . . . . . . . 23
1.3.5 Upper and lower limits. . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.3.6 Stolz-Cesa`ro theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 30
1.3.7 Fixed-point theorems . . . . . . . . . . . . . . . . . . . . . . . . . . 33
1.3.8 Recursions with closed forms . . . . . . . . . . . . . . . . . . . . . . 37
1.3.9 Limits involving the harmonic numbers . . . . . . . . . . . . . . . . 43
1.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2 Infinite Numerical Series 61
2.1 Main Definitions and Basic Convergence Tests . . . . . . . . . . . . . . . . 61
2.2 Raabe and Logarithmic Tests . . . . . . . . . . . . . . . . . . . . . . . . . . 67
2.3 The Kummer, Bertrand, and Gauss Tests . . . . . . . . . . . . . . . . . . . 72
2.4 More Sophisticated Tests Based on Monotonicity . . . . . . . . . . . . . . . 76
2.5 On the Universal Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
2.6 Tests for General Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
2.7 Properties of Convergent Series . . . . . . . . . . . . . . . . . . . . . . . . 84
2.8 Infinite Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
2.9 Worked Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
2.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
3 Continuity 131
3.1 Definition of Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
3.2 Limits of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
3.3 Three Fundamental Theorems . . . . . . . . . . . . . . . . . . . . . . . . . 136
3.4 From the Intermediate Value Theorem to Chaos . . . . . . . . . . . . . . . 147
3.5 Monotone Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
3.6 Worked Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
3.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
4 Differentiation 179
4.1 Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
4.2 Fundamental Theorems of Differentiation . . . . . . . . . . . . . . . . . . . 182
4.3 L’Hˆopital’s Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
vii
viii Contents
4.4 Convex Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
4.5 Taylor’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
4.6 Worked Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
4.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
5 Integration 227
5.1 The Riemann Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
5.2 Classes of Integrable Functions . . . . . . . . . . . . . . . . . . . . . . . . . 232
5.3 The Mean Value Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
5.4 The Fundamental Theorems of Calculus . . . . . . . . . . . . . . . . . . . . 239
5.5 Worked Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
5.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262
6 Sequences and Series of Functions 277
6.1 Pointwise and Uniform Convergence . . . . . . . . . . . . . . . . . . . . . . 277
6.2 Importance of Uniform Convergence . . . . . . . . . . . . . . . . . . . . . . 284
6.3 Two Other Convergence Theorems . . . . . . . . . . . . . . . . . . . . . . . 289
6.4 Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
6.5 Weierstrass’s Approximation Theorem . . . . . . . . . . . . . . . . . . . . . 300
6.6 Worked Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305
6.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328
7 Improper and Parametric Integration 345
7.1 Improper Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345
7.2 Integrals with Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352
7.3 The Gamma Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364
7.4 Worked Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373
7.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399
A List of Problems from MAA 417
Bibliography 421
Index 427
Preface
Mathematics is like looking at a house from different angles.
— Thomas Storer
A conceptually clear induction to fundamental analysis theorems, a tutorial for creative
approaches for solving problems, a collection of modern challenging problems, a pathway
to undergraduate research—all these desires gave life to the pages that follow.
The Mathematical Association of America (MAA) journals have strongly influenced
teaching and research for over a century. They have been an indispensable source that pro-
videsnewproofsofclassicaltheoremsandpublisheschallengingproblems.Thisisespecially
true for The American Mathematical Monthly, which, according to the records on JSTOR,
is the most widely read mathematics journal in the world. Since its inception in 1894, the
Monthly has printed 50 articles on the gamma function or Stirling’s asymptotic formula,
including the magisterial 1959 paper by Phillip J. Davis, winner of the 1963 Chauvenet
prize.Thisisalsoaplacewhereonefindsproofsof“periodthreeimplieschaos.”[67]Mean-
while, its problem section has become the single most challenging and interesting problem
section, unrivaled since SIAM Review canceled its problem section in 1998. Problems from
the Monthly regularly lead to further research. For example, the Erd¨os distance problem
thatservedas thegenesis ofEuclidean Ramseytheory wasposed in1946 and finallysolved
by Guth and Katz in 2010. Recently, Lagarias made a surprising observation: the Riemann
hypothesis is equivalent to the following elementary statement:
(cid:88)
d≤H +ln(H )eHn, where H =1+ 1 +···+ 1, for all n≥1.
n n n 2 n
d|n
A weaker version of this inequality appeared as Monthly Problem 10949.
Unfortunately, these gems are lost among new papers and new problems, and are rele-
gated to the background. Consequently, from time to time it is rewarding to revisit these
marvelous papers and problems. These gems form the foundation of this book. The pur-
pose of this book is to expose students to these stimulating and enlightening proofs and
hard problems of classical analysis mainly published in the Monthly. The goal is two-fold.
First, we present proofs as a form of exploration rather than just a manipulation of sym-
bols. Knowing something is true is far from understanding why it is true and how it is
connected to the rest of what we know. The search for a proof is the first step in the
search for understanding. Drawing on the papers from the MAA journals, numerous proofs
that are conceptually clear have been chosen. I have reexamined the proofs of these many
standard theorems with an eye toward understanding them. Each proof provides either a
novel presentation of a familiar theorem or a lively discussion of a single issue, sometimes
withmultiplederivationsofmanyimportanttheorems.Specialattentionispaidtoinformal
exploration of the essential assumptions, suggestive heuristic considerations, and roots of
the basic concepts and theorems. Second, we collect problems that will promote creative
techniquesforproblem-solvingandundergraduateresearch.Eachproblemisselectedforits
natural charm—the connection with an authentic mathematical experience, its origination
from the ingenious work of professionals, develops well-shaped results of broader interest.
ix